Which of the following are true and which false? Give proofs or counterexamples.I said this was false and my counterexample was .i). If (this is a mapping, I couldn't quite find the right arrow) is continous at c then f is differentiable at c.
This function is continous at 0 but which is not differentiable at 0.
I'm not really sure about this one.ii). If a,b) \rightarrow \mathbb{R}" alt="\exists \ ga,b) \rightarrow \mathbb{R}" /> and a,b) \rightarrow \mathbb{R}" alt="Za,b) \rightarrow \mathbb{R}" /> such that then f is differentiable at c if:
a). Z is bounded
b). g is differentiable at c with g'(c)=0 and
c). g(c)=0;
I think it's true but I don't know how to prove it.
I put false and used the counterexample when c=0.iii). In ii) f is differentiable assuming only (a) and (c).
and .
Once again I put false and used the counterexample where c=0.In ii). f is differentiable assuming only (a) and (b).
Where and
This is false because could be discontinous.In (ii). f is differentiable assuming only (b) and (c).
My counterexample was at c=0.
Here and .
are these correct?